Pythagoras' Theorem allows the length of one side of a triangle to be calculated if the lengths of the other two sides are known. This is especially useful when finding the distance between two points of a coordinate system.
Pythagoras and his theorem
Pythagoras was a Greek mathematician and philosopher who lived over 2500 years ago. He is often credited with discovering the relationship between the length of the sides on a right-angled triangle although this relationship was used by others long before Pythagoras' time.
The right-angled triangle below is shown along with the squares of the length of each of its three sides.
Pythagoras' Theorem is, that for a right-angled triangle, the area of the square on the hypotenuse (the hypotenuse is the longest side in a right-angled
triangle) equals the sum of the areas of the squares on the other two sides . In other words, the square of the longest side equals the squares of the two other sides added together.
There are many different proofs of Pythagoras' Theorem. You will find a very good listing here. Note: You would usually only be expected to recall one of these proofs.
Examples of Solving Problems with Pythagoras' Theorem
What is the length of side c?
|Recall formula ||c2 = a2 + b2
|Substitute in numbers ||c2 = 32 + 42
Note: Since we will be adding a2 and b2 it does not matter which side is "a" and which is "b". Just call one "a" and the other "b".
|Square both numbers ||32 = 3 x 3 = 9
42 = 4 x 4 = 16
|Add them together ||c2 = 9 + 16
|Find the square root (the number that multiplied by itself equals 25) ||
c = √25
c = 5
More Problems Solved With Pythagoras
c2 = 62 + 82
c2 = 36 + 64
c2 = 100
c = √100
c = 10
c2 = 52 + 122
c2 = 25 + 144
c2 = 169
c = √169
c = 13
c2 = 72 + 92
c2 = 49 + 81
c2 = 130
c = √130
c = 11.40
Two points to remember when checking your answers:
- the hypotenuse (the side opposite the right angle) is always the longest of the three sides
- the hypotenuse can never be longer than the length of the other two sides combined.
More problems with other sides unknown
Pythagoras' Theorem can be used to calculate the length of any side of a right-angled triangle if the other two lengths are known.
202 = a2 + 162
202 - 162 = a2 + 162 - 162
400 - 256 = a2
144 = a2
√144 = a
a = 12
122 = 9 2 + b2
122 - 9 2 = 92 - 92 + b2
144 - 81 = b 2
63 = b2
√63 = b
b = 7.94
When the hypotenuse is one of the two known lengths, as in the two examples above, the shorter length is squared and then subtracted from the square of the hypotenuse.
You may have noticed that in some of the examples above, the lengths of each side were all whole numbers. These triangles are known as Pythagorean Triangles. The best known of these is the 3, 4, 5 triangle. There are other triangles that have all three sides with an exact whole number for their lengths. These include triangles with lengths of sides ( 5, 12, 13) , ( 7, 24, 25) , & ( 8, 15, 17).
Pythagoras' Theorem Worksheets
Applying Pythagoras' Theorem
There are examples here showing how Pythagoras' Theorem can be applied to find the distance between two points with given coordinates.